Question

If sin x + cosec x = 2, then write the value of sinⁿ x + cosecⁿ x.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ \sin x+\mathrm{cosec}x=2 \\ \Rightarrow \sin x+\frac{1}{\sin x}=2 \\ \Rightarrow \frac{{\sin }^{2}x+1}{\sin x}=2 \end{gathered}$$

^{$$\begin{gathered} \Rightarrow {\sin }^{2}x+1=2\sin x \\ \Rightarrow {\sin }^{2}x+1-2\sin x=0 \\ \Rightarrow {\left(\sin x-1\right)}^2=0 \\ \Rightarrow \sin x-1=0 \\ \Rightarrow \sin x=1 \\ \mathrm{And},\mathrm{cosec}x=\frac{1}{\sin x}=1 \\ \therefore {\sin }^{n}x+{\mathrm{cosec}}^{n}x=1^n+1^n \\ =1+1=2 \end{gathered}$$}